Chapter 2 Application of Computational Mass Transfer (I) Distillation Process

Transcription

1 Chapter 2 Application of Computational Mass Transfer (I) Distillation Process Abstract In this chapter, the application of computational mass transfer (CMT) method in the forms of two-equation model and Rayleigh mass flux model as developed in previous chapters to the simulation of distillation process is described for tray column and packed column. The simulation of tray column includes the individual tray efficiency and the outlet composition of each tray of an industrial scale column. Methods for estimating various source terms in the model equations are presented and discussed for the implementation of the CMT method. The simulated results are presented and compared with published experimental data. The superiority of using standard Reynolds mass flux model is shown in the detailed prediction of circulating flow contours in the segmental area of the tray. In addition, the capability of using CMT method to predict the tray efficiency with different tray structures for assessment is illustrated. The prediction of tray efficiency for multicomponent system and the bizarre phenomena is also described. For the packed column, both CMT models are used for the simulation of an industrial scale column with success in predicting the axial concentrations and HETP. The influence of fluctuating mass flux is discussed. Keywords Simulation of distillation Tray column Packed column Concentration profile Tray efficiency evaluation Nomenclature A c 1, c 2, c 3 C C C μ, C 1ε, C 2ε, C 3ε c 0 c 02 D D t Surface area per unit volume of packed column, m 1 Model parameters in transport equation for the turbulent mass flux Concentration, kg m 3 Average concentration, kg m 3 Model parameters in k ε model equations Fluctuating concentration, kg m 3 Variance of fluctuating concentration, kg2 m 6 Molecular diffusivity, m 2 s 1 Turbulent mass diffusivity, m 2 s 1 Springer Nature Singapore Pte Ltd K.-T. Yu and X. Yuan, Introduction to Computational Mass Transfer, Heat and Mass Transfer, DOI / _2 51

2 52 2 Application of Computational Mass d e d H d p E o E MV E ML Equivalent diameter of random packing, m Hydraulic diameter of random packing, m Nominal diameter of the packed particle, m Overall efficiency Murphree tray efficiency on gas basis Murphree tray efficiency on liquid basis p ffiffiffiffiffi F F factor, U G q G,ms 1 (kg m 3 ) 0.5 g Acceleration due to gravity, m s 2 G Production term H Height of packed bed measured from column bottom, m h f Height of the liquid layer in tray column, m h w Weir height in tray column, m Overall liquid phase mass transfer coefficient in tray column, K OL ms 1 k Turbulent kinetic energy, m 2 s 2 k G Gas phase mass transfer coefficient in packed column, kg m 2 s 1 k L Liquid phase mass transfer coefficient in packed column, kg m 2 s 1 L Liquid flow rate per unit cross section area, kg m 2 s 1 l w Weir width, m m Distribution coefficient r Position in radial direction, m R Radius of the column, m S C Source term in species conversation equation, kg m 3 s 1 S m Source term in momentum equation, N m 3 t Time, s U Superficial velocities, m s 1 U Interstitial velocity vector, m s 1 u 0 i Fluctuating velocity, m s 1 W Weir length, m x Distance in x direction, m; mole fraction in liquid phase y Distance in y direction, m; mole fraction in gas phase z Distance in z direction, m Z Total height of packed bed, m b L, b V Volume fraction of liquid phase, vapor phase a re Relative volatility ε Turbulent dissipation rate, m 2 s 3 e c 0: Turbulent dissipation rate of concentration fluctuation, kg 2 m 6 s 1 γ Porosity distribution of the random packing bed μ, μ G Liquid and gas phase viscosity, kg m 1 s 1 ρ, ρ G Liquid and gas phase density, kg m 3 σ Surface tension of liquid, N m 1

3 2 Application of Computational Mass 53 σ k, σ ε χ U Correction factor in k ε model equations Characteristic length of packing, m Enhancement factor Subscripts G Gas i Coordinates in different direction; component in solution in Inlet L Liquid 0 Interface b Bulk Distillation is a vapor liquid separation process widely employed in petrochemical, chemical, and allied industries nowadays. The simulation of distillation has long been investigated since 1930s. There are two basic types of distillation equipment: column with tray structure (tray column) and column with packing (packed column). For the tray column, the early approach of simulation is based on the concept of equilibrium tray where the thermodynamic equilibrium between liquid and vapor phases is achieved; and it converts to actual tray by means of empirical tray efficiency. The later advance is to use the rate equation to account for the mass transfer instead of using empirical efficiency and equilibrium relationships. These methods are on the overall basis with the assumption that the flow and concentration are uniform on the column tray. In the 1990s, the application of CFD to a column tray enables to calculate the velocity distribution (velocity profile), yet the calculation of concentration distribution is still lacking. Nevertheless the concentration distribution is even more important and interested by the chemical engineers as it is the deciding factor for predicting the tray efficiency. The recently developed computational mass transfer enables to overcome this insufficiency and provides a rigorous basis for predicting all transport quantities, including the concentration distribution, of a distillation column. The status of packed column simulation is similar to that of tray column. The efficiency of distillation process is very important in optimal design and operation as it is closely related to the column size needed and heat energy consumed. The accurate modeling of distillation process enables to show the nonideal distribution of concentration as well as the fluid flow and the designer and operator can take steps to overcome such nonideality, so as to improve the separation ability of the distillation process.

4 54 2 Application of Computational Mass 2.1 Tray Column The tray column simulation involves mainly the following aspects: Velocity distribution to show the deviation from ideal flow: It can be calculated by using CFD as described in Appendix 1; Concentration distribution for the calculation of tray efficiency: As stated in Appendix 2, the conventional way of using turbulent Schmidt number Sc t model for predicting the concentration distribution is not dependable for the reason that the correct Sc t is not only hard to guess but also it is varying throughout the process. Hence the recently developed c 02 e c0 two-equation model and the Reynolds mass flux model are recommended to use as described in the subsequent sections c 02 e c0 Two-Equation Model Interacted liquid phase form (see Appendix 2.6) of two-equation model is employed in this section for process simulation Model Equations (I) The CFD equation set (k ε model, see Appendix 1) The detailed development of the CFD equations is given in Appendix 1. The equations with a name prifix A refer to those in Appendix 1. Overall mass ðq L b L U Li Þ = S m ða1 3Þ Momentum q L b L U Li U ¼ Lj b L l L b L q L u 0 i u0 j þ b L q L S Li i ða1 4Þ q L u 0 Li u0 Lj ¼ Li Lt Lj j 3 q Ld ij k L ða1 8Þ

5 2.1 Tray Column 55 k L L b L U Li k L L l L þ l Lt i r l Lt b j L b L e L i ða1 11aÞ e L L b L U Li e L L l L þ l i r e C e1 b L e L k L l Lj C e2 b L q L e 2 L k L ða1 13aÞ In foregoing equations, the subscript L denotes the liquid phase. (II) Heat transfer equation set ( T 02 e T 0 model, see Chap. 2): The detailed development of the heat transfer equations is given in Appendix 2. The equations with a name prifix A refer to those in Appendix 2 Energy conservation þ U L T Li ¼ 2 qu 0 Li b C L þ b T0 L þ q L b L S T ða2 3Þ i or written þ U L T i ¼ k b qc L 2 u 0 Li þ b L þ b L S T 2 u 0 Li ¼ ab L þ b L þ b L S T i u 0 Li T0 ¼ a ða2 3aÞ ða2 4Þ T 02 L b L T Lb L U i T 02 b L q 02 a Lt þ a r T 0 2b L q L 2b L q L e T 0 ða2 7aÞ

6 56 2 Application of Computational Mass e T 0 L b L e T Lb L U Li e T 0 a t L b L i r et 0 þ a C T1 b L q L e T 0 T 02 u0 i e 2 T C T2 b L q 0 L T C e L e T 0 T3b 02 L q L k L ða2 10Þ a t equation a Lt ¼ C T0 k L!1 k L T 02 2 ða2 6Þ e L e T 0 Model constant are: C T0 ¼ 0:11, C T1 ¼ 1:8, C T3 ¼ 2:2, C T2 ¼ 0:8, r T 0 ¼ 1:0, r et 0 ¼ 1:0. If the latent heat of vaporization of the component species in distillation process is approximately equal, the conservation equation of energy (heat) can be omitted and the mathematical model comprises with only CFD and mass transfer equation sets. Otherwise, the heat transfer equation set should be involved. (III) Mass transfer equation set ( c 02 e c 0 model, see Chap. 3): Species mass L U Lj C L u 0 Lj c0 þ b L S n i u 0 Lj c0 ¼ j ð1:3þ c 02 L L U Li c 02 L " L D þ D # 02 L i r c 02 2b L D 2 L e c 0 L i ð1:10þ e Lc 0 equation U Li b Lc 0 L i D þ D 0 r ec C c1 b L D t e Lc 0 c 2 2 e 2 Lc C c2b L i c 02 L C c3 b L e Lc 0e Lc 0 k L ð1:17þ

7 2.1 Tray Column 57 D Lt equation D Lt ¼ C c0 k L k L c 02 L e L e Lc 0!1 2 ð1:6þ Model constants are as follows: C c0 ¼ 0:14, C c1 ¼ 1:8, C c2 ¼ 2:2, C c3 ¼ 0:8, r c 2 ¼ 1:0, r ec 0 ¼ 1:0 In the foregoing equations, the fraction of liquid b L in the liquid vapor mixture for tray column can be calculated by the following correlation [1]: " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 0:91 b L ¼ exp 12:55 U G q G q L q G ð2:1þ Usually the net amount of interfacial mass transfer exchange between liquid to vapor and vapor to liquid phases on a tray is small, q L and q G can be considered practically unchanged, so that b L is substantially constant. It should be noted that in the case of packed column, the b L is varying because the porosity of the packing is nonuniformly distributed especially in the near wall region as described in Sect Evaluation of Source Terms The present simulated object is an industrial scale sieve tray column of FRI which is 4 ft. in diameter with six sieve trays for (1) separation of n-heptane and methylcyclohexane [2] and (2) stripping of toluene from dilute water solution [3]. They reported the outlet composition and the tray efficiency of each tray under different operating conditions. The details of this column are given in Fig. 2.1 and Table 2.1 [2]. The operating pressure is 165 kpa. In the numerical computation, the model equations should be first discrete into a large number of small finite elements and solved by algebraic method. Thus, the empirical correlations can be applied to the discrete elements under their local conditions, such as velocity, concentration, and temperature obtained in the course of numerical computation. Note that the local conditions should be within the applicable range of the correlation. Since the latent heat of vaporization and condensation as well as the density of n- heptane and methylcyclohexane are practically equal, the amount of n-heptane transferred from liquid phase to the vapor phase is almost equal to the amount of methylcyclohexane transferred from vapor phase to the liquid phase, thus no material is accumulated or depleted on the tray and the liquid density is kept substantially constant. For this separating system, it can be letting the source term S m ¼ 0.

8 58 2 Application of Computational Mass Fig. 2.1 Structure of simulated sieve tray [4] Table 2.1 Dimension of simulated sieve tray Item Value Item Value Column diameter (m) 1.2 Clearance under downcomer 38 (mm) Tray spacing (mm) 610 Downcomer area (m 2 ) 0.14 Hole diameter and spacing Effective bubbling area (m 2 ) (mm mm) Outlet weir (height length) (mm mm) Hole area (m 2 ) Sun [5 7] and Li [8] simulated this column using interacted liquid phase modeling form with the assumption that the liquid density on a simulated single tray is constant, but for the multi-tray simulation, the density should be changed tray by tray. The source term S Li in the momentum conservation equation can be calculated by one of the following modes: (A) Based on superficial vapor velocity: For the x, y direction [9] S Li ¼ q GU G q L h L U Li ; i ¼ x; y; ð2:2þ

9 2.1 Tray Column 59 where h L is given by [10]: h L ¼ 0:0419 þ 0:189h w 0:0135F s þ 2:45L=W For the z direction [11] ð S Lz ¼ 1 b LÞ 3 U 2 G g(q L q G ) ju G U L j(u G U Lz ) (B) Based on sieve hole vapor velocity As the vapor velocity leaving the sieve holes is much higher than the superficial and sometimes even forming jet flow; such influential effect can not be ignored, especially under the condition of high F-factor. Referring to Fig. 2.2, the three-dimensional vapor velocities leaving the sieve hole can be expressed as follows [12]: " # D h U Gz ¼ 4:0U h z exp d 2 0:1z U Gx ¼ U Gr cos h U Gy ¼ U Gr sin h U Gr ¼ 1 rffiffiffisffiffiffiffiffiffi 3 M 0 1 g g 3 =4 4 p q G z ð1 þ g 2 =4Þ 2 g ¼ r d z ; where M 0 is the momentum of the gas phase flowing out the sieve hole. Fig. 2.2 Coordinate of a sieve [4]

10 60 2 Application of Computational Mass The source term S Li in the momentum equation involves the drag force by the jetting vapor F drag and the resistance F p created by the liquid vapor cross flow. The F drag is given by [13] F drag ¼ C T q G ðu Gi U Li ÞjU G U L j=h f i ¼ x; y; z The F p in the x direction is calculated by [12] F p ¼ C p q L U 2 x =h f; where C p = 0.4; h f ¼ h L b L. The source term S Li is given as follows: S x ¼ F drag þ F p S j ¼ F drag;j K (j = y,z) (C) Comparison between two modes Sun computed the velocity distribution of experimental simulator (Sect for details) by using foregoing two modes to show their difference. The result is shown in Fig As seen from Fig. 2.3, the velocity in mode A is more uniformly distributed than that in mode B except in the region near the column wall. Moreover, the average velocity in the main flow region of mode B is slightly higher than that in mode A but lower locally near the wall. Computation further reveals that, for a large diameter sieve tray with large number of uniformly distributed sieve holes, the simulated results show no substantial difference by using either mode. In subsequent calculation, mode B is used. Fig. 2.3 Liquid velocity profiles obtained by using different modes, operation condition: z = 38 mm, F S = m/s (kg/m 3 ) 0.5, L = m 3 /s, a based on superficial vapor velocity mode, b based on sieve hole vapor velocity [4]

11 2.1 Tray Column 61 The source term S n in the species mass conservation equation represents the component species transferred from one phase to the other, which can be calculated by the conventional mass transfer equation S n ¼ K OL acl C L ; ð2:3þ where K OL (m 2 s 1 ) is the overall mass transfer coefficient; a (m 2 m 3 ) is the effective interfacial vapor liquid contacting area; C L (kg m 3 ) is the average liquid mass concentration in equilibrium with the vapor flowing through the tray; K OL can be given by 1 K OL ¼ 1 k L þ 1 ; mk G ð2:4þ where k L and k G are, respectively, the film coefficients of mass transfer on liquid side and gas (vapor) side, respectively, m is the coefficient of distribution between two phases, which is conventionally called Henry s constant. The value of m is dependent on the concentration of the species concerned. If the concentration change on a tray is not large, the value of m might be taken at the average concentration. However, for the simulation of a multi-tray column, where the change of concentration in the column is appreciable, the value of m should be redetermined for each tray. The k L, k G, and a can be calculated by the empirical equation given by Zuiderweg [14] as follows: k G ¼ 0:13 q 0:065 ð1:0\q G q 2 G \80 kg=m 3 Þ G 1 k L ¼ 0:37 1 mk G ð2:5þ ð2:6þ The effective vapor liquid interfacial area a is calculated by a ¼ a0 h f, where h f is the height of liquid level, a 0 is given by [14]: a 0 ¼ 43 Fbba 2 h 0:53 LFP F 0:3 ; r where F bba is the F factor based on the vapor velocity passing through the bubbling area; h L ¼ 0:6hw 0:5p0:25 b 0:25 ðfpþ 0:25 (25 mm < h w < 100 mm), FP ¼ U L q 0:5, L U G q G b is the weir length per unit bubbling area.

12 62 2 Application of Computational Mass x=0 Fig. 2.4 Diagram of boundary conditions [15] Boundary Conditions Inlet (inlet weir, x = 0): the liquid velocity and concentration are considered as uniformly distributed, so that U L ¼ U L;in, C ¼ C in (Fig. 2.4). For the k ε equations, the conventional boundary conditions are adopted [16]: k in ¼ 0:003Ux;in 2 and e in ¼ 0:09k 3=2 in = 0:03 W 2. The inlet conditions of c 02 e c 0 equations, as presented by Sun [5], are: c 02 L;in ¼ ½ 0:082 ð C C in ÞŠ 2 e Lc0 ;in ¼ 0:9 e L;in k L;in Outlet (outlet weir overflow): ¼ 0. Solid border (tray floor, inlet weir wall outlet weir wall and column wall): the boundary conditions for the mass flux are equal to zero. The wall surface is considered to be no-slip of liquid flow. Interface of the vapor and liquid: we c 02 in ¼ ¼ 0, and U z ¼ Simulated Results and Verification (I) Separation of n- Heptane and Methylcyclohexane The model equations were solved numerically by using the commercial software FLUENT 6.2 with finite volume method. The SIMPLEC algorithm was used to solve the pressure velocity coupling problem in the momentum equations. The

13 2.1 Tray Column 63 Fig. 2.5 C 6 concentration distribution on trays [4] a at 20 mm above the floor of tray number 8, b 70 mm above the floor of tray number 8, c at 20 mm above the floor of tray number 6, d 70 mm above the floor of tray number 6 second-order upwind spatial discretization scheme was employed for all differential equations. Samples of the computed results, Fig. 2.5a, b show, respectively, the computed concentration distribution on tray 8 and tray 6. Unfortunately, no experimental data on the concentration field of the tray is available in the literature for the comparison. However, we may compare indirectly by means of the outlet concentration of each tray. From the concentration distribution on a tray as shown in Fig. 2.5a, b, the outlet composition of each tray can be obtained as shown in Fig. 2.6a and compared with the experimental data. As seen from Fig. 2.6a, the computed outlet concentration of each tray is in good agreement with the experimental measurement except for the tray 4. As we understand, for the total reflux operation, the outlet concentration should form a smooth curve on the plot. The deviation on tray 4 is obvious and likely to be due to experimental error or some other unknown reasons. The average deviation of the outlet composition is 3.77 %.

14 64 2 Application of Computational Mass Cyclohexane concentration (%) (a) Experimental data Two-equation model Tray number. Effiiciency MV, [%] (b) experimental data Two-equation model simulation Tray number Fig. 2.6 Simulation results and experimental data a outlet concentration, b Murphree tray efficiency (reprinted from Ref. [5], Copyright 2005, with permission from American Chemical Society) Another way of comparison is by means of individual tray efficiency. The common expression of tray efficiency is the gas phase Murphree efficiency which is defined by E MV ¼ y n y n þ 1 y n y n þ 1 ð2:7þ where y n is the species concentration (mole fraction) of gas phase in equilibrium with the liquid phase concentration x n (mole fraction); y n and y n+1 are, respectively, the gas concentration in mole fraction leaving and entering the tray. The comparison between simulated results and experimental data is showed in Fig. 2.6b, in which disagreement in tray number 3 and 4 reveals the experimental error in the outlet concentration from tray 4 because the tray efficiency can not be as high as 150 % for tray 3 and as low as 20 % for tray 4. The overall tray efficiency of all trays in the column is commonly used for distillation column evaluation in order to reduce the error of individual tray efficiency. Figure 2.7 shows the simulated overall tray efficiency versus experiment pffiffiffiffiffi measurement under different vapor rate expressed as F factor F ¼ U G q G. The simulation is seen to be confirmed. A feature of computational mass transfer is able to predict the liquid turbulent mass diffusivity D t which is commonly regarded as representing the extent of backmixing (nonideal flow) and thus it is an influential factor to the tray efficiency. Figure 2.8 display the distribution of D Lt on tray number 8 at the depth z of 50 and 100 mm, respectively, apart from tray floor. As seen from the figure, the D Lt is nonuniformly distributed, which reflexes the effectiveness or efficiency of mass transfer is varying with position on a tray. The volume average of D Lt calculated is compared with experimental data reported by Cai and Chen [17] for the same tray column under different vapor rate (F factor) as shown in Fig Although the experimental measurement is

15 2.1 Tray Column Overall tray efficiency E O, % Experimental data Two-equation model F factor Fig. 2.7 Simulated overall tray efficiency and experimental data (reprinted from Ref. [5], Copyright 2005, with permission from American Chemical Society) Fig. 2.8 Distribution of liquid turbulent mass diffusivity on tray number 8, a Tray No. 8, 50 mm above tray floor, p = 165 kpa, L = m 3 /h, b Tray No. 8, 100 mm above tray floor p = 165 kpa, L = m 3 /h [4] performed by using inert tracer technique and the comparison is only approximate, yet it demonstrates that the prediction of D Lt is feasible by using the method of computational mass transfer without doing tedious experimental work Simulated Results and Verification (II) Stripping of Toluene from Dilute Water Solution Kunesh [3] reported the experimental data for the column as shown in Fig. 2.1 for the stripping of toluene from dilute water solution. They gave the outlet composition and the tray efficiency of each tray under different operating conditions.

16 66 2 Application of Computational Mass Fig. 2.9 Simulated turbulent mass diffusivity and experimental data [4] Turbulent mass diffusivity, [m 2 /s] Experimental data Two-equation model F-factor, [m/s(kg/m 3 ) 0.5 ] Sun [7] simulated the outlet concentration of each tray, expressed in area-weighted average, versus tray number for a typical run is shown in Fig and compare with the experimental data. According to the Fenske-Underwood equation under constant relative volatility and low concentration, a plot of logarithmic concentration versus tray number should yields a straight line. In Fig. 2.10, both simulated and experimental points are shown closely to a line with agreement each other. The conventional method of assuming constant turbulent Schmidt number, Sc t, for instance equal to 0.7, is also shown in Fig. 2.10, and the deviation of arbitrary assuming a constant Sc t can be clearly seen. The simulated concentration distribution on a sieve tray is given in Fig. 2.11, in which the stripping action on the tray is seen to be unevenly progressed. Based on the simulated concentration distribution as shown in Fig, 2.11, the local tray efficiencies can be obtained. The simulated tray efficiency by area average Fig Simulated outlet concentration and experimental measurement, Run Q L = 76.3 m 3 /h, Fs = 1.8 (m/s)(kg/m 3 ) 0.5 (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) Toluene outlet concentration, [mole fraction] 1E-4 Experimental data 1E-5 Turbulent Sc=0.7 1E-6 Two-equation model 1E-7 1E-8 1E-9 1E-10 1E-11 1E Tray number

17 2.1 Tray Column 67 Fig Simulated concentration distribution, tray 6, Run Q L = 76.3 m 3 /h, Fs = 1.8 (m/s) (kg/m 3 ) mm above tray floor (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) Fig Simulated and experimental tray efficiency under different m V L Q L = 76.3 m 3 /h, Fs = 1.8 (m/s) (kg/m 3 ) 0.5 (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) Efficiency [%] Experimental data Simulated results mv/l for run is 33.4 % in comparison with the experimental value of 36 %. More simulated tray efficiencies at different mv/l are compared with the experimental measurements as shown in Fig. 2.12, in which reasonable agreement is seen between them. As another example of illustration, the simulated distribution of D Lt across the tray for run is shown in Fig The diverse distribution of D Lt is chiefly due to the complicated non-uniform flow and concentration distributions on the tray. In practice, the mass transfer diffusivity is expressed macroscopically by the volume average. For instance, the predicted volume average values of D Lt for three runs under different operations are 0.035, and m 2 /s, respectively, which are within the reasonable range reported by Cai and Chen [17].

18 68 2 Application of Computational Mass Fig Distribution of turbulent mass transfer diffusivity, tray 6, Run Q L = 76.3 m 3 /h, Fs = 1.8 (m/s) (kg/m 3 ) mm above tray floor (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) Prediction of Tray Efficiency for Different Tray Structures By means of the simulated concentration distribution on a tray, the influence of tray structure on the tray efficiency can be calculated. Sun [7] simulated foregoing sieve tray distillation column as shown in Sect for separating cyclohexane (C 6 ) and n-heptane (n-c 7 ) mixture with different tray structures, including sieve hole arrangement, heights of inlet weir, and outlet weir. As an example of illustration, the tray efficiency with different height of outlet weirs is predicted and compared each other. The simulated concentration distributions on the same sieve tray with different outlet weir height h w are shown in Fig The inlet concentration of C 6 to both trays was in mole fraction; the simulated outlet concentrations for outlet weir height h w equal to 20 and 100 mm were found to be and 0.383, respectively. Higher outlet concentration of C 6 on the h w = 100 tray may be due to deeper liquid layer resulting more interacting area and time between vapor and liquid and therefore enhance the mass transfer. Fig Simulated concentration profile of a sieve trays number 1 at different outlet weir height h w x in = Q L = m 3 h 1, G = 5.75 Kg s 1 P = 165 kpa total reflux 20 mm above tray floor a h w = 20 mm, b h w = 100 mm (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier)

19 2.1 Tray Column 69 Fig Simulated turbulent mass transfer diffusivity profile of sieve trays number 1 x in = 0.482, Q = m 3 h 1, P = 165 kpa, total reflux, 20 mm above tray floor a h w = 20 mm, b h w = 100 mm (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) The corresponding Murphree tray efficiencies obtained were 86.7 and 89.5 % for h w equal to 20 and 100 mm, respectively. The simulated D Lt for both cases are shown in Fig. 2.15, in which different profiles are clearly seen. However, such simulated results do not mean that higher outlet weir is a good choice, as the higher tray efficiency achieved is on the expense of higher pressure drop which means to require more energy of operation. However, it demonstrates that the application of computational mass transfer to evaluate the mass transfer efficiency of different tray structures is feasible, which is helpful in designing new column and assessing existing column Reynolds Mass Flux Model The interacted liquid phase modeling form is employed for present simulation. The simplified assumptions of constant liquid fraction b L and density q L on a tray are applied Standard Reynolds Mass Flux Model Model equations (I) The CFD equation set (k ε model) Overall mass ðq L b L U Li Þ ¼ S m ða1 3Þ

20 70 2 Application of Computational Mass Momentum q L b L U Li U ¼ Lj b L l L b L q L u 0 Li u0 Lj þ S Li i ða1 4Þ where u 0 Li u0 Lj is given L u 0 Li u0 L b L u 0 Li þ U u0 Lj Lk k 0 L L C u0 0 Li k þ l u0 Lj k L k q L b L u Lj Li u0 Lk þ u Li u0 Lk k e L C 1 q L b L u 0 Li k u0 Lj 2 L 3 k Ld ij C2 0 Lb L u 0 Lj Li u0 Lk þ u Li u0 Lk 2 k 3 d ƒƒƒ! iju Li Li u0 Lj j 3 q Lb L e L d ij ða1 23Þ where the constants are: C k = 0.09, C 1 = 2.3, C2 0 = 0.4 [11]. (II) The mass transfer equation set (Reynolds mass flux model) Species mass L U Lj L D L u 0 Lj c0 þ S n i ð1:3þ Fluctuating mass flux u 0 i L u 0 Li LU j u 0 Li j " kl 2 C c1 L b L þ l # 0 Li c 0 j e L q j u e i u0 j C c2 L j k u0 Li c0 þ C c3 b L u Lj c0 j ð1:26aþ where the constants are: C c1 ¼ 0:09, C c2 ¼ 3:2, C c3 ¼ 0:55. Auxiliary equations k L L U Li k L L l L þ l i r Li l Lt b L L b L e L i ða1 11aÞ

21 2.1 Tray Column 71 e L L b L U Li e L L l L þ l i r Le C e1 b L e k l Lj C e2 b L q L e 2 L k L ða1 13aÞ The boundary conditions are the same as given in Sect Verification of simulated result and comparison The simulated column tray is shown in Sect for separating n-heptane and methylcyclohexane. Li et al. [8] and Li [15] simulated the concentration profiles for all trays at different levels from the tray floor, among which the tray number 8 and 6 are shown in Fig. 2.16a, b, respectively. Fig Concentration contour of x y plan on trays by Standard Reynolds mass flux model a 20 mm above tray floor of tray number 8, b 70 mm above tray floor of tray number 8 c 20 mm above tray floor of tray number 6, d 70 mm above tray floor of tray number 6 [15]

22 72 2 Application of Computational Mass Hybrid Reynolds Mass Flux Model The model equations are the same as the Standard Reynolds mass flux model except that the u 0 Li u0 Lj term is simplified by using Eq. (A1-8) as follows: qu 0 Li u0 Lj ¼ l j Lj 2 3 q Ld ij k L ða1 8Þ The hybrid Reynolds mass flux model and algebraic Reynolds mass flux model, which only need to solve simpler Eq. (A1-8) instead of complicated Eq. (A1-23), may be a proper choice for multiple tray computation if their simulated results are very close to the standard Reynolds mass flux model. For comparison, the simulated column trays in Sect for separating n-heptane and methylcyclohexane are used. Li [4] simulated concentration profiles of all trays at different levels above the tray floor, among which the tray number 8 and tray number 6 are shown in Fig. 2.17a, b. Fig Concentration contour of x y plan on trays by Hybrid Reynolds mass flux model, a 20 mm above tray floor of tray number 8, b 70 mm above tray floor of tray number 8, c 20 mm above tray floor of tray number 6, d 70 mm above tray floor of tray number 6 [15]

23 2.1 Tray Column 73 It is found that by comparing Fig. 2.16a versus Fig. 2.17a and Fig. 2.6b versus Fig. 2.17b, the simulated results between standard and hybrid Reynolds mass flux models is practically the same except at the very small region near the end of the inlet weir and the neighboring segmental wall. Such difference is coming from the fact that the standard mass flux model is anisotropic enabling to give more precised three-dimensional flow and mass transfer simulation, while the hybrid Reynolds mass flux model is isotropic and cannot show the detailed three dimensional behaviors in that region. However, if overlooking the difference in this small region, it indicates that the hybrid Reynolds mass flux model can be used for overall simulation instead of using the complicated standard Reynolds mass flux model for the simulation of all trays in a multi-tray column with less computer work. The simulated result by using hybrid Reynolds mass flux model can also compared with that obtained by using two-equation model as shown in Fig. 2.17,in which the agreement between them is seen except in the region near inlet weir and column wall, where the hybrid Reynolds mass flux model gives more detailed concentration distribution than the two-equation model. The verification of hybrid Reynolds mass flux model can also be made by comparing with the experimental outlet concentration of each tray as shown in Figs and 2.19, in which the result by using two-equation model is also presented. It can be seen that the hybrid Reynolds mass flux model gives closer outlet concentration to the experimental measurement than the two-equation model, although both of them are considered to be verified by experiment. The verifications of simulated Murphree tray efficiencies by using hybrid Reynolds mass flux models and two-equation model can also be checked by comparing with experimental data as shown in Figs. 2.19a. The comparison of outlet C 6 from each tray between Standard and hybrid Reynolds mass flux model is given in Fig. 2.19b, in which the agreement between them is seen. Fig Comparison between hybrid Reynolds mass flux model and two-equation model by simulated concentration contours of 20 mm above tray floor on tray number 8 a Hybrid Reynolds mass flux model (reprinted from Ref. [8], Copyright 2011, with permission from Elsevier), b c 02 e c 0 Two-equation model (reprinted from Ref. [5], Copyright 2005, with permission from American Chemical Society)

24 74 2 Application of Computational Mass Fig Comparisons of simulated results by different models with experimental data [15] a tray efficiency, b outlet C 6 concentration. b Comparisons of the outlet C 6 concentration between standard and hybrid Reynolds mass flux model simulations and experimental data [15] Tray efficiency (a) Experimental data Hybrid Reynolds mass flux model Two-equation model Tray number Outlet C 6 concentration from each tray (b) Experimental data Standard RMF model Hybrid RMF model Tray number Generally speaking, the overall simulated result of a distillation tray column by using two-equation model and different Reynolds mass flux model is very close each other and checked with experimental measurements, but if detailed mass transfer and flow information s on the trays are needed, the standard Reynolds mass flux model is the better choice. Reynolds mass flux In this section for convenience, the fluctuating mass flux u 0 Li c0, which is the negative Reynolds mass flux u 0 Li c0, is used for illustration instead of using Reynolds mass flux. In the course of solving the model equation, the fluctuating mass flux u 0 x c0, u 0 y c0, u 0 z c0 can be obtained simultaneously [15]. The radial distributions of them at different axial position of tray 8 are given in Fig. 2.20a. The sum of fluctuating mass flux in all directions, u 0 Li c0 ¼ u 0 x c0 þ u 0 y c0 þ u 0 z c0, is shown in Fig. 2.20b.

25 2.1 Tray Column 75 (a) Inlet weir x=0 m (c) x=0.2 m x=0.4 m x=0.6 m r Outlet weir centerline x=0.8 m (Approximate) (b) Fluctuating mass flux (x direction) (d) x =0.2 m x =0.4 m x =0.6 m r (m) Fluctuating mass flux (y direction) x=0.2 m x=0.4 m x=0.6 m r (m) Fluctuating mass flux (z direction) x=0.2 m x=0.4 m x=0.6 m r (m) (e) Sum of x,y,z dimension fluctuating mass flux / kg m -2 s z=0.2 m z=0.4 m z=0.6 m r /m x axial distance from the inlet weir Fig Simulation results of fluctuating mass flux on tray number 8 by Standard Rayleigh mass flux model [15] a the tray for simulation, b x direction (main flow), c r direction (perpendicular to main flow), d z direction (depth), e profiles of u 0 Li c0 ¼ u 0 x c0 þ u 0 y c0 þ u 0 z c0 at different axial distance of tray As seen in Fig. 2.20b, the fluctuating mass flux u 0 x c0 is greater near the inlet weir region (x = 0.2) than that around the outlet weir region (x = 0.6) because c as well as is c 0 is decreased with x(main flow) direction in distillation process. In the r direction, which is perpendicular to the main flow, all the u 0 x c0, u 0 y c0 and u 0 z c0 contours are almost unchanged up to about 0.3r, then slightly increasing until about r = 0.45r reaching the maximum. This tendency is seen both in Figs. 2.20b d.

26 76 2 Application of Computational Mass Such maximum point indicates the appearance of greatest mass flux transport as well as turbulent diffusion and vortical mixing there due to the impact with the reversed flow (large scale vortex) created in the segmental region of the column. Such simulated result is consistent with many experimental works that the reversed flow was observed around this region. In Figs. 2.19b and 2.20 the u 0 i c0 and u 0 x c0 contours along r (radial) direction showing almost zero gradient from r = 0 to about 0.3 indicates that the turbulent (fluctuating) mass flux flow remains constant, i.e., the turbulent effect is kept steady in this region (see Sect ). However, the foregoing mentioned variation of concentration is very small and cannot be found clearly in the profile of concentration contour Algebraic Reynolds Mass Flux Model The hybrid Reynolds mass flux model can be further reducing the complexity of model equations by setting the convection term on the left side of Eq. (1.26) equal to the turbulent and molecular diffusions term on the right side under steady condition to obtain Eq. (1.27) as shown below. u 0 k i c0 ¼ C c2 cb L e u 0 i u0 þ u c0 i þ C j C c2 e u0 i c0 j ð1:27þ The algebraic Reynolds mass flux model is using Eq. (1.277) to replace Eq. (1.26); all other model equations are the same as the hybrid Reynolds mass flux model. To testify this model, Li [4] simulated the sieve tray column as mentioned in Sect The concentration profile on tray No. 8 are simulated and compared with the simulated results by using different Reynolds mass flux models as shown in Fig. 2.21, from which it is seen that the standard give more detailed information than the other two simplified models although generally speaking their simulated profiles are similar. The comparison can also be made by the outlet concentration and Murphree efficiency of each tray as shown in Figs and The simulated outlet concentrations by using algebraic Reynolds mass flux model are slightly higher than that by the hybrid model; while in Fig the simulated Murphree tray efficiencies by using algebraic Reynolds mass flux mode are slightly lower; although both of them are seen to be sufficiently confirmed by experimental data except tray 4, where experimental error is obvious.

27 2.1 Tray Column 77 Fig Simulated results of the C 6 concentration profiles of the x y plane on tray number 8 simulated by different Reynolds flux models for F = 2.44 m s 1 (kg m 3 ) 0.5 [15] a Standard Reynolds mass flux model at 70 mm above the tray floor, b Standard Reynolds mass flux model at 20 mm above the tray floor, c Hybrid Reynolds mass flux model at 70 mm above the tray floor, d Hybrid Reynolds mass flux model at 20 mm above the tray floor, e Algebraic Reynolds mass flux model at 70 mm above the tray floor, f Algebraic Reynolds mass flux model at 20 mm above the tray floor

28 78 2 Application of Computational Mass Fig Comparison of simulated results for outlet concentrations of each tray by different models with experimental data [15] OutletC 6 concentration Experimental data Standard RMF model Hybrid RMF model Algebraic RMF model Tray number Fig Comparison of simulated results for tray efficiencies by different models with experimental data [15] Murphree tray efficiency Experimental data Standard RMF model Hybrid RMF model Algebraic RMF model Tray number Prediction of Multicomponent Point Efficiency Difference Between Binary and Multicomponent Point Efficiency The separation efficiency in multicomponent distillation is quite different with that in binary (two components) distillation in the following aspects: (1) In binary system, the diffusion flux between liquid and vapor phases is proportional to the negative concentration gradient; while it is not true in multicomponent system. (2) In binary system, the diffusion coefficient for component i and j is equal; while in multicomponent it is not equal.

29 2.1 Tray Column 79 (3) In binary system, the range of point efficiency is from 0 to 1; while it is ranging from to + for multicomponent system. The complication appeared in multicomponent system is chiefly due to the complex nature of molecular interaction to form nonideal solution and may appear bizarre behaviors (see Sect ). The point efficiency is an essential information in distillation design and operation. The tray efficiency can be calculated by the CMT models presented in this chapter, it shows that the tray efficiency is in connection with the tray structure, flow pattern and operating conditions and thus it is only referred to a specific distillation column under specific condition. On the other hand, the point efficiency which depends on only the local condition of vapor liquid contact and the physical properties of the system is the better way to evaluate the feasibility of using distillation tray column for the separation. The research on point efficiency has been undertaken over many decades and developed different expressions under the name of the author, like Murphree [18], Hanson [19], Standart [20] and Holland [21]. Among them, the Murohree point efficiency E MV is commonly used, which is defined for tray column as the ratio of the concentration decrease of vapor between entering and leaving the tray and that if the leaving vapor is in thermodynamic equilibrium with the liquid on the tray. Mathematically it can be expressed as follows: E MV ¼ y i;n y i;n þ 1 y i;n y ; ð2:8þ i;n þ 1 where subscript n denotes the tray number; y n þ 1 and y n are, respectively, the concentration (component i) of vapor entering and leaving the tray; y i;n is the vapor concentration in equilibrium with the liquid at this local point. Note that the subscript i and j in this section refer to the component i and j, respectively, but not the coordinate direction of flow. The nomenclature can be seen clearly from Fig Fig Micro-element (cell) taken on sieve tray

30 80 2 Application of Computational Mass Fig Formation of vapor bubbles from sieve holes in different zones Murphree point efficiency can be also expressed in terms of liquid concentration as follows: E ML ¼ x i;n 1 x i;n x i;n 1 x i;n ð2:9þ The vapor phase Murphree point efficient E MV is frequently used especially in distillation, while the liquid phase point efficient E ML is suitable for the liquid phase control processes, such as absorption and desorption processes. Precisely, the mass transfer undertaken in the vertical direction above the tray deck is complicated as shown typically in Fig. 2.25, involving jetting, dispersed bubbles, splashing as well as the generation of liquid drops as entrainment in the tray spacing. Usually it is divided into three zones, i.e., froth zone (jetting), bubble dispersing zone (free bubbling), bubble breaking zone (liquid drops splashing as entrainment in tray space). Since the bubble breaking (splash) zone has very small contribution to the mass transfer, the first two zones, in which the liquid as continuous phase and the vapor as dispersed phase, are dominant and have been established as two-zone model in the literature The Oldershaw Sieve Tray The sieve tray developed by Oldershaw [22, 23] is recognized as the common distillation tray to be used for representing the point efficiency. The construction parameters are given in Table 2.2 and the column is shown in Fig The simulation of which is the convenient way to find the point efficiency of the corresponding separating system. Wang [24] simulated the Oldershaw sieve tray [24] with consideration of using two zones model for the liquid on the tray. The distillation is three components nonideal solution (ethanol, isopropanol, water) for the purpose of investigating the bizarre phenomenon of multicomponent distillation.

31 2.1 Tray Column 81 Table 2.2 Main construction parameters of Oldershaw sieve tray Parameter Value Tray diameter (mm) 38 Diameter of tray spacing (mm) 64 Sieve hole diameter (mm) 1.25 Thickness of tray floor (mm) 1.2 Perforation (%) 6.38 Height of outlet weir (mm) Fig Construction and operation of Oldershaw sieve tray (reprinted from Ref. [23], Copyright 1987, with permission from American Chemical Society) For the nonideal multicomponent vapor liquid system, the Maxwell Stefan equation is usually employed to evaluate the mass transfer behaviors. The fundamentals of Maxwell Stefan equation is briefly introduced in Sect Experimental Work on Multicomponent Tray Efficiency Wang [24] performed the following experimental works to verify the simulation. Experiment was conducted in Oldershaw sieve tray as shown in Fig Two multicomponent systems are used for testing the point efficiency, i.e., three-component system (ethanol, isopropanol, water) and four-component system (ethanol, isopropanol, tert-butyl alcohol, water). The initial composition of three component system in sequence is as follows: (x b ) ¼ ð0:447 7,0:220 9,0:331 4Þ T The composition of entering vapor is (y F ) ¼ ð0:444 7,0:221 4,0:333 9Þ T

32 82 2 Application of Computational Mass Fig Experimental setup (1 column, 2 Oldershaw tray, 3 downcomer, 4 Reboiler, 5 heating pot, 6 flow meter, 7 reflux tube, 8 cooling water meter, 9 condenser, P pressure measuring point, T temperature measuring point, S sampling point) The operating conditions are: temperature T = K, Q V = m 3 s 1, h L = mm. The experimental setup is shown schematically in Fig Simulation Model for Point Efficiency For calculating Murphree tray efficiency, we need to know the composition of vapor leaving the tray y out which can be obtained as follows. Since the range of composition change on a tray is small, we may assume the vapor liquid equilibrium relationship to be linear, i.e., ðy Þ¼½K eq Š½CŠðx b Þ; ð2:10þ where ½K eq Š is (n 1) rank diagonal matrix, representing the equilibrium constant of the binary pairs. Also at the interface, ðy 0 Þ¼[K eq Š½CŠðx 0 Þ; ð2:11þ where y 0 is the vapor concentration at interface. Then we have

33 2.1 Tray Column 83 ðy y 0 Þ¼½K eq Š½CŠðx b x 0 Þ The mass flux transferred can be calculated by (see Sect of Chap. 3) N L i N V i ¼ c t b L R L 1½C L Šðx b x 0 Þ ð1:47þ ¼ c t b V R V 1 C V ðxb x 0 Þ ð1:48þ In the calculation, the liquid bulk concentration x b is known. Equation set (1.47) and (1.48) can be solved by stepwise iteration as given below to obtain the mass flux being transferred between liquid and vapor phases N i N i ¼ Ni L ¼ Ni V. For the vapor passing through the liquid on the tray, the vapor concentration is changing from y in to y out, and should be calculated by differential method. Take a differential element Dh on the sieve tray as shown in Fig. 2.28, we have and dg i ¼ N i a Adh dg i ¼ c V t u sa Ady; where G i is the vapor flow rate; A is the area of the bubbling zone on the tray; a is the surface area of the bubbles. Combine foregoing equations to yield dy i ¼ N ia c V dh t au s Integrating consecutively above equation from h = 0 at the tray deck to h = h 1 for the bubble formation zone and from h 1 to h 2 for the dispersed bubble zone, the y out can be found for calculating the point efficiency. The trial and error method for Fig Differential element on the tray

34 84 2 Application of Computational Mass stepwise calculation is employed to obtain the solution. The equations needed for computation of each zone are given below. Bubble formation zone Experimental work shows that the main form of vapor in this layer is jetting. The diameter of the vapor jet d j which is related to the liquid height h L and the diameter of sieve hole d h, was correlated by Hai [25]: d j ¼ 1:1d h þ 0:25h L Thus the surface area of the jet column is as follows: a ¼ 4ud j ðd h Þ 2 ; where u is the fraction of hole area. As the vapor flow through the jet column is similar to its flow through the falling film column, the mass transfer coefficient k V can be calculated by the following relationship for two-component system [26]: sffiffiffiffiffiffiffi k V D ¼ 2:0 V ¼ 0:046 DV ðreþ 0:96 ðscþ 0:44 pt V d j Re ¼ d ju j q V ; Sc ¼ l V l V q V D V u j ¼ Q V ½ðp=4ÞðdÞ 2 Šu d 2 h d h ¼ u h d j d j 2 ; t V ¼ h j ; u j where d j is the diameter of the vapor jet; u j and u h are, respectively, the vapor velocity based on jet diameter and sieve hole diameter. Bubble dispersion zone The vapor column reaching to this zone is broken into bubbles of different size and distributed diversely. The average diameter of the bubble can be estimated by the following equation [27]: d max ¼ ð0:5we c Þ 0:6 r We c ¼ sd max r q L 1=3 ; qv q L 0:6 ðu s gþ 0:4 q V q L 0:2 where We c is Weber group; r is the surface tension; s is the residence time which is given by [27] s ¼ 2q L ðu s gd max Þ 4=3

35 2.1 Tray Column 85 It was reported [28, 29] that the ratio of average and maximum bubble diameters is a constant, i.e., d av d max ¼ 0:62 The reliability of foregoing estimation is seen to be confirmed by some experimental data from literature as shown in Table 2.3. The vapor fraction b V in this layer for sieve hole smaller than 2 mm can be estimated by equation below b V 1 b V ¼ 8:5Fr 0:5 ; Fr 4: u 0:56 b V 1 b V ¼ 1:25u 0:14 Fr 0:25 ; Fr [ 4: u 0:56 ð Fr ¼ u sþ 2 ; gh L where u is the fraction of sieve perforation on the tray. By the iteration of foregoing equations, the d av can be obtained as well as the surface area of the bubble by a ¼ 6 d av b V The mass transfer coefficient between bubble and the liquid on the tray was measured for binary system by Zaritzky [31] and correlated by Prado and Fair [32] as follows: k V ¼ Sh DV d av Sh ¼ 11:878 þ 25:879ðlgPeÞ 5:640ðlgPeÞ 2 Q V Pe ¼ d avu b D V ; u b ¼ ; p 4 ðdþ2 b V q V Table 2.3 Calculated bubble diameter compared with experimental measurements Sieve hole (m/s) Calculated d av (mm) Experimental value by Sharma [29] (mm) Experimental value by Raper [30] (mm) Experimental value by Geary [27] (mm)

36 86 2 Application of Computational Mass Fig Vapor column from sieve hole where D V is the molecular diffusivity of component i in the vapor phase. Steps of calculation As seen in Fig. 2.29, let the height of the two liquid zones on the tray be h ðh ¼ h 1 þ h 2 Þ, take a differential element Dh on the tray where y in ¼ y h and y out ¼ y h þ RDh. The mass flux of component i in the element can be calculated as follows: 1. Let y in ¼ y bh and assume y out ¼ y 0 bh þ Dh, the average concentration of component i is y av ¼ 1 2 y bh þ y bh þ Dh 2. Calculate the mass flux to be transferred by aforementioned method so as to obtain the concentration of vapor leaving from the differential element. If it is close enough to the assumed value, then proceed to the next differential element above until reaching to the top of the liquid zone to obtain the outlet vapor concentration from the tray. As an example, Wang [24] give the calculated result along liquid height h as shown in Table 2.4 As seen, the mass transfer is high at low liquid level and decrease as the vapor goes up to the top of the froth. It indicates the bubble formation zone is dominant in the mass transfer process Simulated Results and Comparison with Experimental Data The comparisons between simulated and experimental Murphree point efficiencies of three-component and four-components systems are given, respectively, in Tables 2.5 and 2.6. The error in most cases is less than few percent which is acceptable for estimation purpose.

37 2.1 Tray Column 87 Table 2.4 Calculated result of mass flux transferred along liquid height Liquid height h from tray deck (mm) Vapor concentration, mole Mass flux transferred N (mol m 2 s 1 ) fraction Ethanol Isopropanol Water Ethanol Isopropanol Water Table 2.5 Comparison of simulated point efficiency with experimental data (I) [system: ethanol (1), isopropanol (2), water (3)] Expt. No. Component Liquid concentration on tray, mole fraction Experimental point efficiency Simulated point efficiency Error = Sim. Exp

38 88 2 Application of Computational Mass Table 2.6 Comparison of simulated point efficiency with experimental data (II) [system: ethanol (1), isopropanol (2), tert-butyl alcohol (3), water(4)] Expt. No. Component Liquid concentration on tray, mole fraction Point efficiency Experimental Simulated Error The Bizarre Phenomena of Multicomponent System The bizarre phenomena of multicomponent system can be illustrated by the case of three component system as calculated by Wang given in preceding section. The simulated diffusion flux of isopropanol is plotted versus driving force of mass transfer ðy 0 yþ as shown in Fig

39 2.1 Tray Column 89 Fig The diffusion mass flux of isopropanol in three components system versus driving force of mass transfer As seen in the figure that the driving force ðy 0 yþ is positive between A and B, the direction of mass transfer is from y 0 (vapor) to y (liquid). At point B, although the driving force is positive, but the mass flux of isopropanol transferred is zero; such phenomenon is regarded as diffusion barrier which is not happened in binary system. From point B to C, the driving force is still positive, yet the isopropanol transferred is negative, i.e., the direction of mass transfer is reversed and such phenomenon is regarded as reversed diffusion. Moreover, as seen in Fig. 2.31, at the liquid height about h = 25, the driving force is approaching zero, but the isopropanol still able to undertake mass transfer between phases; such phenomenon is regarded as osmotic diffusion. It should be mentioned that such bizarre phenomena is only happened for isopropanol in the three component system but not for ethanol and water. Thus, the complication of nonideal multicomponent system depends on many factors and still under investigation. The plot of simulated results is also given in Fig Fig Mass transfer flux and driving force of isopropanol in three components system versus liquid height

40 90 2 Application of Computational Mass 2.2 Packed Column The simulation of packed column by computational mass transfer methodology have been made by Liu [33] and Li [8] as given in following sections:. The model assumptions are the same as the tray column except that axially symmetrical condition is applied for the packed column. The packed column to be simulated are that reported by Sakata [2], it is 1.22 m in diameter packed with 50.8 mm carbon steel Pall ring of 3.66 m height for separating n-heptane and methylcyclohexane under kpa and total reflux operation c 02 e c Two-Equation Model Modeling Equations The model equation for packed column, comprised CFD equation set and mass transfer equation set. Unlike the tray column, the porosity of packed column is nonuniformly distributed and the liquid fraction b L should be retained in the model equations. The interacted liquid phase model equations are. Overall mass ðq L cb L U Li Þ ¼ S m ; ð2:12þ where c is the porosity of the packed bed. Momentum L cb L U Li @U Li ¼ cb L þ L l L q L u 0 Li u0 Lj i þ cb L S Li ð2:13þ where u 0 Li u0 Lj by q L u 0 Li u0 Lj ¼ l j Lj 2 3 q Ld ij k L ða1:8þ k L L cb L U Li k L L l L þ l Lt i r kl j þ cb L ðg Lk e L Þ ð2:14þ

41 2.2 Packed Column 91 e L L cb L U Li e L L l L þ l i r el þ cb L ðc 1e G Lk C 2e q L e L Þ e L k L ð2:15þ The model constants are c l = 0.09, σ k = 1.0, σ ε = 1.3, C 1e = 1.44, C 2e = Species mass conservation equation c 02 L U Li c L U Li L D Lt u 0 Li c0 þ S n i " L D L þ D # 02 i r c 02 2cb L D Lt 2 L e c 0 ð1:10þ i e c 0 c 0 U i cb L L D L þ D 0 i r ec e c 2 e 2 c C c1 cb L D Lt C 0 c2 c 2 c C e c0e c0 c3cb 02 L k L ð1:17þ D t equation!1 k L c D Lt ¼ C c0 k 02 2 L ð1:6þ e L e c 0 Model constants are: C c0 ¼ 0:14, C c1 ¼ 1:8, C c2 ¼ 2:2, C c3 ¼ 0:8, r c 2 ¼ 1: Boundary Conditions Inlet (reflux at column top x = 0): U L ¼ U in, V L ¼ 0, C i ¼ C i;in. For the other parameters, we may set to be [1, 34]: k L;in ¼ 0:003U 2 L;in e L;in ¼ 0:09 k1:5 L;in d H c 02 ¼ð0:082C in Þ 2 ¼ 0:0067Cin 2 e c 0 ¼ 0:4 e in c 02 in k in

42 92 2 Application of Computational Mass Outlet (column bottom): fully developed turbulent condition is assumed so that the gradients of all parameters U except pressure are set to ¼ 0 Column symmetrical axis (r = 0): the radial gradients of all parameters U except pressure are equal ¼ 0 Column wall (r = R): the relevant flux is equal to zero. Near column wall region: standard wall function is employed Evaluation of Source Term As stated in Sect. 2.1, considering the latent heat of both species is almost equal, so that S n ¼ 0. The source term S Li is expressed by S Li ¼ q L g þ F LS;i þ F LG ; where F LS is the flow resistance created by random packing, F LG is the interface drag force between liquid and vapor phases. The F LS can be evaluated by using following correlation [35]: " # ð1 cþ 2 ð1 cþ F LS ¼ Al L c 2 de 2 þ Bq L ju L j U L ; cd e where U is interstitial velocity vector; c is the porosity; d e is the equivalent diameter of the packing; constants A = 150, B = The F LG is calculated by F LG ¼ Dp L U slip; U slip where Dp L is the wet-bed pressure drop; U slip is slip velocity vector between vapor and liquid and equal to U slip ¼ U G U L The S n in species equation, similar to the tray column, can be calculated by:

43 2.2 Packed Column 93 S n ¼ K OL acg C L 1 K OL ¼ 1 k L þ 1 mk G The gas and liquid film coefficients k L, k G and the volumetric effective surface area a are obtained from the correlation by Wagner et al. [34] as follows: k L ¼ 4U LD L U 0:5 L phcv k G ¼ 4U GD G U 0:5 G ; pc ð h cþv where the enhancement factor Φ L and Φ G is set equal to 1 under experimental condition; χ is characteristic length depending on bed height Z: v ¼ C 2 pk Z The coefficient C pk for 50.8 mm pall ring packing is equal to The vapor liquid contacting area a is calculated by [34] a ¼ hc a T 1:0 c ; where a T is the specific area of the packing; c is the porosity; h is the total liquid holdup of the packing which comprises static holdup h s and dynamic holdup h d. For 50.8 mm Pall ring packing, h s is calculated by [36] h s ¼ 0:033 exp 0:22 gq L r L a 2 T and h d by [37] h d ¼ 0:555 a TUL 2 1=3 gc 4: Simulated Result and Verification Separation of Methylcyclohexane and n-heptane Average axial concentration along column height and verification The simulated radial averaged axial concentration along radial direction at different column height as shown in Fig The plot is made by ln x 1 x versus column height z (z is the height of the packed bed measured from the column

44 94 2 Application of Computational Mass C 6, ln(x/(1-x)) (a) F-factor=0.758 m s -1 (kg m -3 ) Bottom H (m) Top C 6, ln(x/(1-x)) (b) F-factor=1.02 m s -1 (kg m -3 ) Bottom H (m) Top (c) C 6, ln(x/(1-x)) F-factor=1.52 m s -1 (kg m -3 ) Bottom H (m) Top Fig Comparisons of the concentration profiles in liquid phase between two-equation model predictions (solid lines) and experimental data (circles) (H is height of packed) a F- factor = m s 1 (kg m 3 ) 0.5, b F-factor = 1.02 m s 1 (kg m 3 ) 0.5, c F-factor = 1.52 m s 1 (kg m 3 ) 0.5 (reprinted from Ref. [33], Copyright 2009, with permission from Elsevier) bottom) because according to the Fenske equation such plot should be in a straight line at constant relative volatility which is applicable to the present case. The simulated curve is nearly a straight line and in good agreement with the experimental data. HETP and verification The separation efficiency of packed column is usually expressed in terms of HETP (Height Equivalent of Theoretical Plate). According to the Fenske equation, the slope of ln x 1 x versus Z plot is equal to ln a re HETP where a re is the relative volatility of the separating system. The simulated HETP can be obtained from Fig by smoothing the computed curve to a straight line and find the slop. As shown in Fig. 2.33, the simulated HETP is confirmed by the experimental data. Turbulent mass diffusivity distribution The volume average turbulent mass diffusivity D Lt computed by the two-equation model is shown in Fig at different F factor, and more detailed distribution is given in Fig These figures show that the turbulent mass diffusivity is higher in the upper part of the column and lower in the near wall region. The reason is due to higher concentration around the upper column in distillation

45 2.2 Packed Column Experimental data Two-equation model HETP (m) F-factor (m/s(kg/m 3 ) 0.5 ) Fig HETP comparison between predictions and measurements (reprinted from Ref. [33], Copyright 2009, with permission from Elsevier) 0.10 F-factor=0.758 m s -1 (kg m -3 ) 0.5 D Lt,av (m 2 s -1 ) F-factor=1.02 m s -1 (kg m -3 ) 0.5 F-factor=1.52 m s -1 (kg m -3 ) Bottom H (m) top Fig Average turbulent mass diffusivity along the column height at different F factor (reprinted from Ref. [33], Copyright 2009, with permission from Elsevier) process so as to undertaking more quantity of mass transfer. At the same time the wall effect accounts for the mass transfer lower down in the near wall region Reynolds Mass Flux Model Li [8] simulated the packed column as described in Sect by using Reynolds mass flux model instead of two-equation model and compare their difference. The simulated results for three forms of Reynolds mass flux model (standard, hybrid and algebraic) are given in subsequent sections.

46 96 2 Application of Computational Mass D Lt (m 2 s -1 ) H=3.0m H=2.6m H=2.2m H=1.8m H=1.0m H=1.4m H=0.6m H=0.2m r/r Fig Distribution of turbulent mass diffusivity in the column F = 1.02 m s 1 (kg m 3 ) 0.5, H Height of packed bed (H = 0 at the column bottom) (reprinted from Ref. [33], Copyright 2009, with permission from Elsevier) Standard Reynolds Mass Flux Model Interacted liquid phase model with constant fluid density q and constant liquid fraction b is employed for simulation. The model equations are. Overall mass L U Li ¼ S m ð2:17þ Momentum L U Li U Lj ¼ j cb L l Li qu 0 Li u0 Lj þ cb L S Li ð2:18þ where u 0 Li u0 Lj is calculated 0 Li u0 þ U 0 Li u0 k C 2 0 C k e u0 i u0 Li u0 Lj j þ k q u 0 Li u0 Lk u 0 Li u0 k þ u 0 Lj u0 0 Li u0 k e C 1 u 0 Li k u0 Lj 2 3 k Ld ij Li k 3 e Ld ij j þ u Li u0 Lk 2 k 3 d iju 0 Li u0 Lk where the constants are: C k = 0.09, C 1 = 2.3, C 2 = 0.4.

47 2.2 Packed Column 97 Species mass conservation L U Li C Fluctuating mass flux 0 Li Lju 0 Li j where C c1 ¼ 0:09, C c2 ¼ 3:2, C c3 ¼ 0:55. Auxiliary equations k L L U Li k L e L L cb L U Li e L D L u 0 Li c0 þ cb L S Ln i " k L C c1 u 0 j e u0 Lj þ l 0 Li c 0 L j u e Li u0 Lj C j k u0 Li c0 þ C c3 u Lj c0 j L l L þ l i r Li l t cb L L cb L e L i L l L þ l i r e C e1 cb L e L k L l Lj C e2 cb L q L e 2 L k L ð1:3þ ð1:26aþ ða1:11aþ ða1:13aþ The model constants are c l = 0.09, σ k = 1.0, σ ε = 1.3, C 1e = 1.44, C 2e = The boundary conditions and the evaluation of source terms are the same as given in Sects and Simulated result and verification The simulated C 6 concentrations profile of the whole column at different F factor is shown in Fig In comparison with Fig simulated by using two-equation model, the concentration in the main flow area is almost the same but in the near wall region is somewhat difference. The volume average axial concentration distribution is given in Fig. 2.37, in which the simulated curve is seen to be in agreement with the experimental data. Reynolds mass flux The fluctuating mass flux (negative Reynolds mass flux) in axial and radial directions and their sum are given in Figs and In the distillation column tray, the species concentration is decreasing from inlet to the outlet weir, i.e., under negative gradient. The positive u 0 x c0 means that the diffusion of turbulent mass flux u 0 x c0 is consistent with the bulk mass flow and promotes the mass transfer in x direction.

48 98 2 Application of Computational Mass Fig Concentration profile of C 6 by standard Reynolds mass flux model, a F = m s 1 (kg m 3 ) 0.5, b F = 1.02 m s 1 (kg m 3 ) 0.5, c F = 1.52 m s 1 (kg m 3 ) 0.5 [15] C 6, ln(x A /(1-x A )) (a) F-factor=0.758 m s -1 (kg m -3 ) 0.5 Standard Reynolds mass flux model Experimental data Bottom H (m) Top C 6, ln(x A /(1-x A )) (b) F-factor=1.02 m s -1 (kg m -3 ) 0.5 Experimental data Standard Reynolds mass flux model Bottom H (m) Top (c) C 6, ln(x A /(1-x A )) F-factor=1.52 m s -1 (kg m -3 ) 0.5 Experimental data Standard Reynolds mass flux model Bottom H (m) Top Fig Average C 6 concentration along column height at different F factors [15]

49 2.2 Packed Column 99 (a) Flectuating mass flux in x direction/ kg m -2 s H=3.0 m H=2.7 m H=2.3 m H=1.9 m H=1.5 m H=1.1 m r/r (b) Fluctuating mass flux in y direction/ kg m -2 s H=2.7 m H=2.3 m H=1.9 m H=1.5 m r/r Fig Simulated fluctuating mass flux in Axial (x) and radial (y) directions at different bed height H a u 0 x c0, b u 0 y c0 [15] Fig Profiles of u 0 x c0 þ u 0 y c0 at different packed height H (H is measured from column bottom) [15] Sum of fluctuating mass flux in two direction/ kg m -2 s H=2.7 m H=2.3 m H=1.9 m H=1.5 m r/r As seen in Fig. 2.38a, most of the u 0 Li c0 gradient in y (radial) directions is almost zero around the column centerline (r/r = 0) of the lower part of the column (H < 1.9 m) indicating only molecular diffusion is existed. At the upper part of the column (H > 2.3 m), u 0 x c0 contour is increasing from r/r = 0 to about r/r = 0.7, indicating the turbulent diffusion u 0 x c0 is promoted with increasing rate (see Sect ). Afterward, from r/r = 0.7, the slope is turning to negative, which means the diffusion rate is decreasing until about r/r = Thus, the diffusion of u 0 x c0 in radial direction displies wavy changes and follows the pattern of decreasing increasing decreasing increasing sharply decreasing sharply near the column wall. In Fig. 2.38b, the u 0 y c0 contours behave similar to the u 0 x c0 indicating the radial u 0 y c0 diffusion is variating with the pattern of decreasing increasing decreasing sharply to the column wall. As seen in Fig. 2.39, the overall tendency of u 0 Li c0 (equal to u 0 x c0 þ u 0 y c0 ) is similar to both u 0 x c0 and u 0 y c0. It is noted that u 0 x c0 is much greater than u 0 y c0 in this case, that means the u 0 Li c0 diffusion is dominated by u 0 x c0.

50 100 2 Application of Computational Mass It should be noted that the radial variation in concentration is small and may not be seen clearly in the concentration profile of the whole column. However, the detailed information about the mass transfer, which can be obtained by using Rayleigh mass flux model, is helpful to the column design and the evaluation of process efficiency Hybrid Reynolds Mass Flux Model The model equations are the same as the standard Reynolds mass flux model except that the calculation of u 0 Li u0 Lj is by Eq. (A1.8) instead of Eq. (A1.23). Simulated result and verification The simulated C 6 concentration profile of whole column is shown in Fig. 2.40, which is almost identical with Fig The simulated radial averaged axial concentration distribution is compared with experimental data and the simulated result by using standard Reynolds mass flux model as shown in Fig These figures display no substantial different between hybrid and standard Reynolds mass flux models. The comparison of simulated result on radial averaged axial concentration between hybrid Reynolds mass flux model and two-equation model is given in Fig As seen from the figures, both show close to the experimental data and the one better than the other only in upper or lower part of the column. The simulated HETP by hybrid Reynolds model is compared with that by two-equation model as shown in Fig The prediction by hybrid Reynolds model is better than two-equation model for low and high F factors but not in the intermediate range. Fig Concentration profiles by hybrid Reynolds mass flux model a F = m s 1 (kg m 3 ) 0.5, b F = 1.02 m s 1 (kg m 3 ) 0.5, c F = 1.52 m s 1 (kg m 3 ) 0.5 (reprinted from Ref. [8], Copyright 2011, with permission from Elsevier)

51 , 2.2 Packed Column 101 ln(x/(1-x)) C 6 (a) Experimental data Standard Reynolds mass flux model Hybrid Reynolds mass flux model bottom H/ (m) top C 6, ln(x/(1-x) ) (b) Experimental ata Standard Reynolds mass flux model Hybrid Reynolds mass flux model bottom H (m) top C 6, ln(x/(1-x) ) (c) 1.2 Experimental data Standard Reynolds mass flux model Hybrid Reynolds mass 0.6 flux model bottom z (m) top Fig Comparison between standard and hybrid Reynolds mass flux models with experimental data [15] a F = m s 1 (kg m 3 ) 0.5, b F = 1.02 m s 1 (kg m 3 ) 0.5, c F = 1.52 m s 1 (kg m 3 ) Algebraic Reynolds Mass Flux Model The model equations are the same as the standard Reynolds mass flux except u 0 Li u0 Lj and u 0 Li c0 equations are changed to the following algebraic form: u 0 Li u0 Lj ¼ k C 1 e k LC 2 C 1 e L u 0 Lj u0 Lk u 0 Li u0 k þ u 0 Lj u0 k þ u i Lj u0 Lk k 3 u0 Li u0 Lk d k þ 2 1 k L d ij 3 C 1 ða1:24þ where C k = 0.09, C 1 = 2.3, C 2 = 0.4 u 0 Li c0 ¼ k C c2 e u 0 Li u0 þ u c0 Li þ C c3k L u Li j C c2 e c0 Li j ð1:27þ where C 2 = 3.2, C 3 = 0.55.

52 102 2 Application of Computational Mass C 6, ln(x/(1-x) ) (a) Experimental data Hybrid Reynolds mass flux model Two-equation model bottom H (m) top C 6, ln(x/(1-x) ) (b) Experimental data Hybrid Reynolds mass flux model Two-equation model bottom H (m) top C 6, ln(x/(1-x) ) (c) Experimental data Hybrid Reynolds mass flux model Two-equation model bottom H (m) top Fig Comparison of hybrid Reynolds model and two-equation model with experimental data a F = m s 1 (kg m 3 ), b F = 1.02 m s 1 (kg m 3 ), c F = 1.52 m s 1 (kg m 3 ) (reprinted from Ref. [8], Copyright 2011, with permission from Elsevier) Fig HETP by hybrid Reynolds mass flux model and two-equation model (reprinted from Ref. [8], Copyright 2011, with permission from Elsevier) HETP m Experimental data Hybrid Reynolds mass flux model Simulation by two-equation model F-factor ms -1 (kgm -3 ) 0.5

53 2.2 Packed Column 103 Fig Concentration profiles by using algebraic Reynolds mass flux model a F = m s 1 (kg m 3 ) 0.5, b F = 1.02 m s 1 (kg m 3 ) 0.5, c F = 1.52 m s 1 (kg m 3 ) 0.5 [15] The simulated C 6 concentration profiles of the whole column are shown in Fig. 2.44, which is substantially identical with Fig by hybrid Reynolds mass flux model simulation. The verification of algebraic Reynolds mass flux model as well as the comparison with hybrid model is shown in Fig At low F factor, these two models are in agreement with experiment, but at high F factor the algebraic Reynolds mass flux model shows greater deviation from the experimental data. 2.3 Separation of Benzene and Thiophene by Extractive Distillation Extractive distillation is frequently employed for the separation of mixture with close boiling point. It features by adding an extractive agent to increase the relative volatility of the mixture concerned so as to make the separation easier with less number of theoretical plates or transfer unit required. Liu et al. [38] employed this process for the separation of benzene (boiling point C) and thiophene (boiling point C) in a packed column with N-methyl-2pyrrolidone (NMP) as the extractive agent. The flow sheet is shown schematically in Fig The extractive column was 0.19 m in diameter, packed with 2 2 mm stainless h rings packing. The column consisted four sections of 700, 600, 1000, 4000 mm packing, respectively, in sequence from the column top. The operating pressure was kpa. The extractive agent, N-methyl-2pyrrolidone (NMP) was introduced at the column top at 2.4 ml per min. and the feed containing 90 % benzene and 10 % NMP was entered between Sects. 2.2 and 2.3 at 0.4 ml per min.